Stable and Efficient Networks with Farsighted Players: The Largest
Consistent Set
1
Anindya Bhattacharya
Department of Economics,
The University of York,
United Kingdom.
This Version: March, 2005
1
Acknowledgment: To add.
1
Abstract
In this paper we study strategic formation of bilateral networks with
farsighted players in the framework of Jackson and Wolinsky (1996). We
use the largest consistent set (LCS)(Chwe (1994)) as the solution concept
for stability. We show that there exists a value function such that for every
component balanced and anonymous allocation rule, the corresponding LCS
does not contain any strongly efficient network. Using Pareto efficiency, a
weaker concept of efficiency, we find that there exists a value function and a
component balanced allocation rule such that the corresponding LCS does
not contain any Pareto-efficient network. This implies that the well-known
incompatibility between stability and efficiency of networks may persist even
when the players are farsighted. Next we study some possibilities of resolving
this incompatibility.
JEL Classification: C71, D20.
1
Introduction
A network is a representation of relations among agents/players in a society or an
economy. Formally, a network is a graph which describes the structure of association among the agents. The agents are usually represented by the nodes of the
graph and an edge between two nodes represents the relation between two agents.
This relation can be unilateral or bilateral. The corresponding structures of relationship are represented by directed or non-directed networks respectively.
A network is a very powerful tool for describing the structure of association
among agents as a rich pattern of cooperation among them can be captured in this
framework. This framework is also capable of describing pay-off-externalities to
a group owing to the formation of other groups. During the last few years, these
attractive features have stimulated a spate of research in this area of studying
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strategic network formation.2
This study is in the framework of bilateral networks. A player’s action is to form
link(s) with other players or to sever existing link(s). In this set-up, if a link is to
be formed between two players, then consent from both the players is necessary although a player can break a link unilaterally. The now canonical model for analysis
in this framework was introduced by Jackson and Wolinsky (1996). In their model,
the total pay-off to the players when a network is formed is represented by a value
function which assigns a real value to every network. This value is distributed to
the players according to some allocation rule. For analyzing strategic formation of
networks, they introduced the notion of pairwise stability. One central result in
their work showed that the set of stable networks (with respect to the notion of
pairwise stability) and strongly efficient networks, those which are socially optimal,
may be disjoint if the allocation rules are intuitively nice.
This impossibility result was followed by a number of studies further exploring
this incompatibility between socially optimal states and stable states. Two important works where this incompatibility has been sought to be resolved are Dutta
and Mutuswami (1997) and Currarini and Morelli (2000) (both reprinted in Dutta
and Jackson ed. (2003)). However, Dutta and Mutuswami studied the strong and
coalition-proof Nash equilibria of a network formation game. Therefore, either
only myopic coalitional deviations or only internal subcoalitional deviations are
considered in their work. Currarini and Morelli do not consider coalitional moves
and nor are the moves endogenous in their framework.
One important issue is to study the strategic formation of networks and the
relation between stable and efficient networks when the players are farsighted and
the coalitional moves are endogenous. Players are said to be farsighted if they
anticipate that any action by a group of players may generate a further chain of
2
See the book edited by Dutta and Jackson (2003) for a collection of representative literature
and also the survey by Jackson contained therein.
3
actions by some other groups. They take this fact into account when computing
the final pay-off resulting from their moves (in such models perfect information is
assumed). Recently, Dutta and Jackson (2003) have emphasized the need for such
analysis.
“Perhaps the most important issue regarding modeling the formation of network is to develop fuller models of networks forming over
time, and in particular allowing players who are farsighted. Farsightedness would imply that players’ decisions on whether to form a network
are not based solely on current pay-offs, but also where they expect
the process to go and possibly from emerging steady states or cycles in
network formation. ... It is conceivable that, at least in some contexts,
farsightedness may help in ensuring the efficiency of the stable state.”
(Dutta and Jackson in the “Introduction” of Dutta and Jackson (ed.) (2003), emphasis in the original).
The present work is one attempt in that direction. Two notable related works in
this area are Dutta et al. (2003) and Page et al. (2002).
The largest consistent set (LCS) (Chwe (1994)) is possibly the most popular
solution concept for farsighted coalitional stability. It has been observed that the
LCS may be too inclusive and therefore, several refinements of this solution have
been proposed (e.g., Xue (1998), Mauleon and Vannetelbosch (2003) etc.).
However, we find that in spite of the largeness of the LCS, there exists a value
function such that for every component balanced and anonymous allocation rule,
the largest consistent set (with respect to the value function and the allocation rule)
does not contain any strongly efficient network. This impossibility result implies
that the well-known incompatibility between stability and efficiency of networks
persists even when the players are farsighted. We also show that there exists a value
4
function and a component balanced allocation rule such that the largest consistent
set (with respect to the value function and the allocation rule) does not contain
any Pareto efficient network. Next, we study some possibilities of resolving this
incompatibility.
Section 2 gives the preliminary definitions. The results are collected in Section
3. The proof of one of the propositions is given in the appendix.
2
Notation and the Preliminary Definitions
The framework and basic tools for the present analysis were introduced by Jackson
and Wolinsky (1996). Below we give only the essential definitions for completeness.
For an elaborate explanation of these concepts and a number of economic examples that fit into this framework, we refer to the comprehensive survey by Jackson
(2003).
Networks
Let N be a finite set of players. Given S ⊆ N, by g S we denote the set of all
doubleton subsets of S. A bilateral network g on N is a subset of g N . The set of
all possible bilateral networks on N is denoted by Z. Given a non-empty network
g ∈ Z, an element {i, j} ∈ g (where i, j ∈ N ) is a link between players i and j in
the network g. We shall often denote the link between i and j simply by ij. The
empty network (i.e., the network with no links) is denoted by ∅.
Players i and j have an indirect link between them in a network g if there exist
i0 , i1 , . . . , im in N such that i0 = i, im = j and for k = 0, . . . , m − 1, ik ik+1 ∈ g.
Conventionally it is assumed that there is a link between each player and itself. A
network g induces a partition Π(g) of N where two players i and j are in the same
element in the partition if and only if there exists an indirect link between them.
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Given a network g and i ∈ N, by Πi (g) we denote the unique element in Π(g) that
contains the player i. The components of a network g,
C(g) = {g(S)|S ∈ Π(g), |S| ≥ 2}
where g(S) = g S ∩ g.
Therefore, for any network g, g = ∪{g 0 | g 0 is a component of g}.
Throughout this paper we denote the coalitions of players by S, T etc. and the
networks by a, b, g, g 0 etc..
Value Functions and Allocation Rules
Given a network g, a value function v : Z 7→ R assigns a real value to g. This value
is generated by some underlying socio-economic process. We normalize v so that
v(∅) = 0. The set of all such value functions is denoted by V. Given a network g,
let C(g) be the set of the components of g. A value function is component-additive
if for every g ∈ Z,
v(g) =
X
v(g 0 ).
g 0 ∈C(g)
Given a value function v ∈ V, an allocation rule Y : Z × V 7→ RN allocates the
value of a network to the players. Given a value function v ∈ V, an allocation rule
Y : Z × V 7→ RN induces a corresponding preference ordering ºi (v, Y ) for each
i ∈ N on Z given as follows:
for g, g 0 ∈ Z, g ºi (v, Y )g 0 iff Yi (g, v) ≥ Yi (g 0 , v).
An allocation rule is component balanced if for any component additive value function v, g ∈ Z and g 0 ∈ C(g),
X
Yi (g, v) = v(g 0 ),
i∈N (g 0 )
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where N (g 0 ) is the set of players linked in the component g 0 of g. Given a permuta−1
tion π : N 7→ N, let v π be defined by v π (g) = v(g π ) for each g ∈ Z. An allocation
rule Y is anonymous if for every v ∈ V, g ∈ Z and permutation π,
Yπ(i) (g π , v π ) = Yi (g, v)
for each i ∈ N.
We introduce two allocation rules which will be useful later. An allocation rule
Y E is said to be egalitarian if for every v ∈ V and g ∈ Z, YiE (g, v) = v(g)/|N |.
Note that Y E is anonymous but not component balanced. Given any component
additive v ∈ V, the component-wise egalitarian allocation rule Y CE is defined by
YiCE (g, v) =
v(Ci )
|Si |
where Si ∈ Π(g) is the unique partition element containing player i and Ci is the
component of g in which player i is linked. Y CE splits the value equally if the value
function is not component additive. Note that Y CE is component balanced as well
as anonymous.
Efficient Networks
Given a value function v, a network g ∈ Z is strongly efficient if v(g) ≥ v(g 0 ) for
all g 0 ∈ Z.
A network g ∈ Z is Pareto efficient relative to a value function v and an allocation
rule Y if there does not exist g 0 ∈ Z such that Yi (g 0 , v) ≥ Yi (g, v) for all i ∈ N
with strict inequality for some i.
The Environment of Networks
An environment of social networks is represented by G = (N, Z, {ºi }i∈N , {→S
7
}S⊆N ). Here ºi is the preference relation for i ∈ N on Z (induced by some underlying value function and allocation rule). For each i ∈ N, aºi b means that player
i weakly prefers network a to network b. The strict part of ºi is denoted by Âi .
The relation →S is the enforcement relation for S ⊆ N . For any a, b ∈ Z, a→S b
implies that the coalition S can enforce network b from network a. Formally,
Definition 1 (Jackson and van den Nouweland (2001)) A coalition S can enforce
a network b from a network a if and only if
(i) a link ij ∈ b \ a implies that {i, j} ⊆ S and
(ii) a link ij ∈ a \ b implies that {i, j} ∩ S 6= ∅.
For some coalition S and a, b ∈ Z, if a Âi b for all i ∈ S then that is written
as a ÂS b.
Indirect Domination and the Largest Consistent Set (LCS)
Below we give the definitions only. For the motivation of these concepts we refer
to Chwe (1994).
Definition 2 (Chwe (1994)) For a, b ∈ Z, b indirectly dominates a, denoted as
b À a, if there exist a0 , a1 , . . . , am in Z and coalitions S0 , S1 , . . . , Sm−1 such
that a0 = a and am = b and for j = 0, . . . , m − 1,
(i) aj →Sj aj+1 ,
(ii) am ÂSj aj .
Definition 3 (Chwe (1994)) A set Y ⊆ Z is said to be consistent if Y = {a ∈
Z| ∀(S, d) ∈ (2N × Z) for which a →S d, ∃ e ∈ Y such that [e = d or e À d]
and e 6ÂS a}. The set L ⊆ Z is said to be the largest consistent set (LCS) if it is
consistent and it contains every consistent set.
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By Proposition 2 in Chwe (1994), a non-empty LCS exists for every environment
of networks.
3
The Results
First we show an impossibility result which may be seen as an analogue of the
impossibility result of Jackson and Wolinsky (1996) in the environment with farsighted players.
Proposition 1 There exists a value function such that for every component balanced and anonymous allocation rule, the largest consistent set (with respect to
the value function and the allocation rule) does not contain any strongly efficient
network.
Proof: The proof is given in the appendix.
Remark 1 Note that if we drop component balance as a requirement, then for
every value function, every strongly efficient network is in the LCS with respect to
the egalitarian allocation rule Y E . In the next proposition we discuss the implication of dropping anonymity.
By V̄ we denote the class of value functions defined as follows: V̄ = {v ∈
V |v(g) > 0 if and only if g is not totally disconnected}. Dutta and Mutuswami
(1997) studied this class of value functions for their possibility result.
Proposition 2 There exists a component balanced allocation rule such that for
every v ∈ V̄ , the largest consistent set (with respect to the value function and the
allocation rule) contains at least one strongly efficient network.
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Proof: Take a component additive value function v ∈ V̄ . Fix g ∈ Z such that g
is strongly efficient with respect to v. We shall define a component balanced (but
non-anonymous) allocation rule Y such that {g} is internally consistent (see the
Lemma in the appendix) with respect to Y. Then, by the Lemma in the appendix
we are done.
Case 1: Let there exist k ∈ N such that k has no link in g with any other player.
By the definition of V̄ and component additivity, there exists at most one such k.
Take g 0 ∈ Z \ {g}.
Subcase (a): If k is linked in some component hk ∈ C(g 0 ) then Yk (g 0 , v) = v(hk ) and
for every other player j ∈ Πk (g 0 ), Yj (g 0 , v) = 0. For every h ∈ C(g 0 ) \ {hk } fix some
ih ∈ N and set Yih (g 0 , v) = v(h). For every j ∈ N \({k}∪{ih ∈ N |h ∈ C(g 0 )\{hk }}),
Yj (g 0 , v) = 0.
Subcase (b): If k is not linked with any other player in g 0 then for every h ∈ C(g 0 )
fix some ih ∈ N and set Yih (g 0 , v) = v(h). For every j ∈ N \ {ih ∈ N |h ∈ C(g 0 )},
Yj (g 0 , v) = 0.
Case 2: Suppose g is such that every player is linked in g with at least one other
player. Take g 0 ∈ Z \ {g}. For every h ∈ C(g 0 ) fix some ih ∈ N and set Yih (g 0 , v) =
v(h). For every j ∈ N \ {ih ∈ N |h ∈ C(g 0 )}, Yj (g 0 , v) = 0.
And in both these cases, Yj (g, v) = YjCE (g, v). So, every player who is linked with
some other player in g gets a positive pay-off under Y.
Now take S ⊆ N and g 0 ∈ Z such that g 0 6= g and g→S g 0 . Suppose we are in Case 1,
Subcase (a). Note that then S 6= {k}. Let T = N \({k}∪{ih ∈ N |h ∈ C(g 0 )\{hk }}).
Then g indirectly dominates g 0 according to the following sequence of coalitional
moves:
g 0 →T ∅→N \{k} g.
Now suppose we are in Case 1, Subcase (b). Then, again, S 6= {k}. Let T =
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N \ ({k} ∪ {ih ∈ N |h ∈ C(g 0 )}. Then, again, g indirectly dominates g 0 according
to the following sequence of coalitional moves:
g 0 →T ∅→N \{k} g.
Thus, for Case 1, {g} is internally consistent.
Similarly, it can be shown that for Case 2 also, {g} is internally consistent.
An immediate and interesting open question (in the spirit of Dutta and Mutuswami (1997)) is whether we can find a component balanced allocation rule such
that every strongly efficient network is in the largest consistent set.
Now, note that the set of Pareto efficient networks with respect to an allocation
rule is larger than the set of strongly efficient networks. Even then, the LCS (which
is usually quite inclusive) and the set of Pareto efficient networks may be disjoint.
And, for this impossibility result we do not even need anonymity explicitly.
Proposition 3 There exists a value function and a component balanced allocation
rule such that the largest consistent set (with respect to the value function and the
allocation rule) does not contain any Pareto efficient network.
Proof: Take the following environment which is a slight modification of the one
given in the proof of Theorem 1 in Jackson and Wolinsky (1996).
N = {1, 2, 3}.
For notational convenience we partition Z into the subsets C1 to C4 such that:
C1 = {{12, 23, 13}};
C2 = {g ∈ Z| g = {ij, jk}; i, j, k ∈ N };
C3 = {g ∈ Z| g = {ij}; i, j ∈ N };
C4 = {∅}.
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Take the following value function:
v({12, 23, 13}) = 0;
for every g ∈ C2, v(g) = 1 + ²; where 0 < ² < 0.5.
for every i, j ∈ N, v({ij}) = 1,
v(∅) = 0.
Fix the component balanced and anonymous allocation rule Y as follows.
Y1 ({12, 23, 13}, v) = Y2 ({12, 23, 13}, v) = Y3 ({12, 23, 13}, v) = 0,
for i, j, k ∈ N, Yi ({ij, jk}, v) = Yk ({ij, jk}, v) = 0.5, Yj ({ij, jk}, v) = ²;
for i, j, k ∈ N, Yi ({ij}, v) = Yj ({ij}, v) = 0.5; Yk ({ij}, v) = 0,
Yi (∅, v) = Yj (∅, v) = Yk (∅, v) = 0.
Note that given the value function and the allocation rule, the set of Pareto efficient networks is C2. However, routine calculation (see Chwe (1994)) yields that
the LCS for this environment is C3.
We do not know as yet whether this impossibility result can be extended for
any arbitrary component balanced and anonymous allocation rule.
Next, we give a sufficient condition on the value functions which ensures that
there exists an allocation rule for which a strongly efficient network is in the LCS.
Definition 4 (Jackson and van den Nouweland (2001)) A value function v ∈
V is top-convex if some efficient network also maximizes per-capita value among
individuals. Formally, let for coalition S, p(v, S) = maxg∈gS v(g)/|S|. The value
function is top-convex if p(v, N ) ≥ p(v, S) for each coalition S.
We refer to Jackson and van den Nouweland (2001) for a discussion of topconvexity.
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Proposition 4 Suppose a value function v is top-convex. Then every strongly
efficient network is in the LCS with respect to the component-wise egalitarian allocation rule Y CE .
Proof: If v is not component additive, then by the definition of Y CE and Remark
1 we are done.
Suppose v is component additive and let g be a strongly efficient non-empty network. Then per-capita value of every component of g is p(v, N ) (by Jackson and
van den Nouweland (2001), section 4). Note that under Y CE the maximum pay-off
that any i ∈ N can get in any network in Z is p(v, N ). Now suppose g is not in the
LCS. Then, by Proposition 2 in Chwe (1994) there exists g 0 in the LCS such that
g 0 À g. Also note that by top convexity, every i ∈ N is linked with some j 6= i in
g. But then there cannot exist g 0 such that g 0 À g.
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Concluding Remarks
The impossibility results presented in this work reinforce the tension between stability and efficiency in the environment of bilateral networks, first highlighted by
Jackson and Wolinsky. Dutta et al. (2003) get a similar impossibility result using
a different concept of stability with farsighted players.
Of course, this work is far from complete and there are many obvious open
questions (about which we have remarked in the body of the paper) unanswered.
13
References
[1] Chwe, M. S.-Y. (1994): “Farsighted Coalitional Stability”, Journal of Economic Theory, 63, 299-325.
[2] Currarini, S. and M. Morelli (2000): “Network Formation with Sequential
Demands”, Review of Economic Design, 5, 229-250.
[3] Dutta, B., S. Ghosal and D. Ray (2003): “Farsighted Network Formation”,
mimeo. Forthcoming in Journal of Economic Theory.
[4] Dutta, B. and M. O. Jackson (ed.) (2003): Networks and Groups: Models of
Strategic Formation, Springer.
[5] Dutta, B. and S. Mutuswami (1997): “Stable Networks”, Journal of Economic
Theory, 76, 322-344.
[6] Jackson, M. O. (2003): “A Survey of Models of Network Formation: Stability and Efficiency”, mimeo. Available as: http://www.hss.caltech.edu/∼
jacksonm/netsurv.pdf
[7] Jackson and van den Nouweland (2001): “Strongly Stable Networks”, mimeo.
Forthcoming in Games and Economic Behavior.
[8] Jackson, M. and A. Wolinsky (1996): “A Strategic Model of Social and Economic Networks”, Journal of Economic Theory, 71, 44-74.
[9] Mauleon, A. and V. Vannetelbosch (2003): “Farsightedness and Cautiousness
in Coalition Formation”. mimeo.
[10] Page F.H., W. Wooders and S. Kamat (2002): “Networks and Farsighted
Stability”. Forthcoming in Journal of Economic Theory.
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[11] Xue, L. (1998): “Coalitional stability under perfect foresight”, Economic Theory, 11, 603-627.
5
Appendix: Proof of Proposition 1
Before proceeding to the main body of the proof, for later use we note the following
fact in the form of a lemma.
Lemma Call a set Y ⊆ Z internally consistent if a ∈ Y implies the following:
∀(S, d) ∈ (2N × Z) for which a →S d, ∃ e ∈ Y such that [e = d or e À d] and
e 6ÂS a. If Y ⊆ Z is internally consistent then Y ⊆ L.
Proof of the lemma: (from Chwe (1994)) Let Y ⊆ Z be internally consistent.
Define Λ := ∪{X ⊆ Z| X is internally consistent}. To prove the lemma, it suffices
to show that Λ is consistent. To prove this we need to show that a ∈ Z \ Λ implies
that there exists (S, d) ∈ (2N × Z) for which a →S d and for every e ∈ Λ such that
[e = d or e À d], e ÂS a. Suppose not, i.e., let there exist a ∈ Z \ Λ for which the
following is true:
∀(S, d) ∈ (2N ×Z) for which a →S d, ∃ e ∈ Λ such that [e = d or e À d] and e 6 ÂS a.
Then clearly, Λ ∪ {a} is internally consistent which violates the definition of Λ. So,
Y ⊆ Λ ⊆ L.
Now we proceed to the main body of the proof.
Proof of the Proposition 1: Take the following environment which is a slight
modification of the one given in the proof of Theorem 2 in Dutta, Ghosal and Ray
(2003).
N = {1, 2, 3}. For notational convenience later, we partition Z into the subsets
15
C1 to C4 such that:
C1 = {{12, 23, 13}};
C2 = {g ∈ Z| g = {ij, jk}; i, j, k ∈ N };
C3 = {g ∈ Z| g = {ij}; i, j ∈ N };
C4 = {∅}.
Take the following value function:
v({12, 23, 13}) = 9;
for every g ∈ C2, v(g) = 0;
for every i, j ∈ N, v({ij}) = 8,
v(∅) = 0.
Fix any component balanced and anonymous allocation rule Y. Then, by component balance and anonymity,
Y1 ({12, 23, 13}, v) = Y2 ({12, 23, 13}, v) = Y3 ({12, 23, 13}, v) = 3,
for i, j, k ∈ N, Yi ({ij, jk}, v) = Yk ({ij, jk}, v) = c, Yj ({ij, jk}, v) = −2c, where c
is some real number;
for i, j, k ∈ N, Yi ({ij}, v) = Yj ({ij}, v) = 4; Yk ({ij}, v) = 0,
Yi (∅, v) = Yj (∅, v) = Yk (∅, v) = 0.
Here the unique strongly efficient network is {12, 23, 13}. However, below we show
that whatever the value of c, the LCS, L = {{12}, {23}, {13}}. We consider the
following three cases.
Case 1: c ≥ 4 :
First, we show that the set C3 is internally consistent. Therefore, we are to show
that a ∈ C3 implies the following: ∀(S, d) ∈ (2N × Z) for which a →S d, ∃ e ∈ C3
such that [e = d or e À d] and e 6ÂS a. Take x ∈ C3 and let x = {ij}, i, j ∈ N.
Consider (S, d) ∈ (2N × Z) such that x →S d, d 6= x. Then, by Definition 1,
S ∩ {i, j} 6= ∅. If d ∈ C3, then set e = d. If d ∈ C1 ∪ C4 then consider the
enforcement d →{i,j} x and set e = x. Suppose d ∈ C2. Then d is either {lm, mk}
16
or {lk, km} where l, m ∈ {i, j}, l 6= m, k ∈ N \ {i, j}. In the former subcase
consider the enforcement d →{m} x and set e = x. In the latter subcase, consider
the enforcement d →{k} {lk} and set e = {lk}. Since x ºi y and x ºj y for every
y ∈ C3, we are done. Thus, we show that C3 is internally consistent and so, by
the lemma, C3 ⊆ L.
Next we prove that in fact, C3 = L. Suppose not and let some L ⊃ C3 be the
LCS. First, we claim that L ∩ C2 = ∅. Take some x (= {ij, jk}) ∈ C2, i, j, k ∈ N.
Consider the enforcement relation x →{j} {ij}. Then {ij}Âj x. Moreover, since
yÂj x for every y ∈ Z \ {x}, it follows that there does not exist any e ∈ L such that
[e = {ij} or e À {ij} and e 6Âj x]. Thus, the claim is proved. Next, consider x
from C1 ∪ C4 and the enforcement relation x →{1,2} {12}. Note that for any (S, y)
∈ (2N × Z) such that y 6= {12}, {12} →S y implies that S ∩ {1, 2} 6= ∅. Moreover,
for every e ∈ Z \ C2, {12} º1 e and {12} º2 e. Therefore, there does not exist
e ∈ L such that e À {12}. Since {12}Â{1,2} x, x ∈
/ L.
Case 2: −2 < c < 4 :
Note that in this case, for i, j ∈ N, {ij}Â{i,j} g for every g ∈ Z \ C3 and also, for
any g ∈ Z, {ij} ºi g and {ij} ºj g. In this case also, C3 is internally consistent,
i.e., a ∈ C3 implies the following: ∀(S, d) ∈ (2N × Z) for which a →S d, ∃ e ∈ C3
such that [e = d or e À d] and e 6ÂS a. For proving this, take x ∈ C3 and let
x = {ij}, i, j ∈ N. Consider (S, d) ∈ (2N × Z) such that x →S d and d 6= x.
By Definition 1, S ∩ {i, j} 6= ∅. If d ∈ C3, then set e = d. If d ∈ Z \ C3 then
consider the enforcement d →{i,j} x and set e = x. Since x ºi y and x ºj y for
every y ∈ C3, we are done. Therefore, by the lemma, C3 ⊆ L.
In this case also, L = C3. To see this, take x ∈ Z \ C3 and consider the enforcement relation x →{1,2} {12}. Note that for any (S, y) ∈ (2N × Z), y 6= {12},
{12} →S y implies that S ∩ {1, 2} 6= ∅. As noted above, for every e ∈ Z, {12} º1 e
and {12} º2 e. Therefore, there does not exist e ∈ L such that e À {12}. Since
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{12}Â{1,2} x, x ∈
/ L.
Case 3: c ≤ −2 :
Again, first we show that C3 is internally consistent, i.e., we show that a ∈ C3
implies the following: ∀(S, d) ∈ (2N × Z) for which a →S d, ∃ e ∈ C3 such that
[e = d or e À d] and e 6ÂS a. Take x ∈ C3 and let x = {ij}, i, j ∈ N. Consider
(S, d) ∈ (2N × Z) such that x →S d and d 6= x. By Definition 1, S ∩ {i, j} 6= ∅. If
d ∈ C3, then set e = d. If d ∈ C1∪C4 then consider the enforcement d →{i,j} x and
set e = x. Suppose d ∈ C2. Then, d is either {lm, mk} or {lk, km} where l, m ∈
{i, j}, l 6= m and k ∈ N \ {i, j}. In the former subcase consider the enforcement
d →{l,k} {lk} and set e = {lk}. In the latter subcase, consider the enforcement
d →{i,j} {ij} and set e = {ij}. Since x ºi y and x ºj y for every y ∈ C3, we are
done. Therefore, C3 is internally consistent and so, C3 ⊆ L.
Next we prove that once again, in this case also, L = C3. Suppose not and let
some L ⊃ C3 be the LCS. To begin with, we claim that L ∩ C2 = ∅. Take some
x (= {ij, jk}) ∈ C2, i, j, k ∈ N. Consider the enforcement relation x →{i,k} {ik}.
Suppose, there exists e ∈ L such that e À {ik}. We show below that this is
impossible. Let, if possible, a sequence of enforcements by which this indirect
domination occurs be the following:
{ik}(= a1 ) →S1 a2 →S2 . . . →Sm−1 am →Sm e,
where for each l ∈ {1, . . . , m}, Sl is a coalition, al is a network and a2 6= a1 without
loss of generality. Then, by the definition of enforcement relation, S1 ∩ {i, k} 6= ∅.
We consider two subcases. First take the subcase where c < −2. Since S1 ∩{i, k} 6=
∅ and eÂS1 {ik}, e must be either {ji, ik} or {jk, ki}. Therefore, by the definition
of enforcement relation, j ∈ Sl for some l ∈ {1, . . . , m} and by the definition
of indirect domination, eÂSl al . But this is impossible because, for every g ∈ Z,
g ºj {ji, ik} and g ºj {jk, ki}. Next, take the subcase where c = −2. Then,
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{ik} ºi g and {ik} ºk g for every g ∈ Z. Therefore, for this subcase also, there
cannot exist e ∈ Z such that e À {ik}. Since {ik}Â{i,k} {ij, jk}, {ij, jk} ∈
/ L.
Thus, the claim is proved.
Next, take any x ∈ C1∪C4 and consider the enforcement relation x →{1,2} {12}.
Note that for any (S, y) ∈ (2N × Z) such that y 6= {12}, {12} →S y implies that
S ∩ {1, 2} 6= ∅. By the claim above, L ∩ C2 = ∅ and for every e ∈ Z \ C2, {12} º1 e
and {12} º2 e. Therefore, there does not exist e ∈ L such that e À {12}. Since
{12}Â{1,2} x, x ∈
/ L.
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